Abstract: Of the branches of mathematics, geometry has, from the earliest Hellenic
period, been given a curious position that straddles empirical and exact
science.Its standing as an
empirical and approximate science stems from the practical pursuits of artistic
drafting, land surveying and measuring in general.From the prominence of visual applications, such as
figures and constructions in the twentieth century Einsteinís General Theory
of Relativity holds that the geometry of space-time is dependent upon physical
quantities.On the other hand,earlier on in history, the symmetry and perfect regularity of certain
geometric figures were taken as representative of a higher orderknowledge than that afforded by sense experience.Concerns with figures and constructions, instead of with numbers and
computations, rendered geometry amenable to axiomatic formulation and
syllogistic deduction,establishing
a paradigm of demonstrative visual and intuitive knowledge that has spanned two
millennia.

In geometry and as followed in geometrical art there remains a connection
that distinguishes between the unboundedness of spaces as a property of its
extent, and special cases of infinite measure over which distance would be taken
is dependent upon particular curvature of lines and spaces.The curvature of a surface could be defined in terms only of properties
dependent solely on the surface itself as being intrinsic.On the empirical side, Euclidean and non-Euclidean geometries
particularly Riemannís approach effected the understanding of the relationship
between geometry and space, in that it stated the question whether space is
curved or not.Gauss never
published his revolutionary ideas on non-Euclidean geometry, and Bolyai and
Lobachevsky are usually credited for their independent discovery of hyperbolic
geometry.Hyperbolic geometry is
often called Lobachevskian geometry, perhaps because Lobachevskyís work went
deeper than Bolyaiís.However, in
the decades that followed these discoveries Lobachevskyís work met with rather
vicious attacks.The decisive
figure in the acceptance of non-Euclidean geometry was Beltrami.In 1868, he discovered that hyperbolic geometry could be given a concrete
interpretation, via differential geometry.For most purposes, differential geometry is the study of curved surfaces
by way of ideas from calculus.Geometries
had thus played a part in the emergence and articulation of relativity theory,
especially differential geometry.Withinthe range of mathematical properties these principles could be expressed.Philosophically, geometries stress the hypothetical nature of
axiomatizing, contrasting a usual view of mathematical theories as true in some
unclear sense. Steadily over the last hundred years the honor of visual
reasoning in mathematics has been dishonored.Although the great mathematicians have been oblivious to these fashions
the geometer in art has picked up the gauntlet on behalf of geometry.So,metageometry is intended
to be in line with the hypothetical character of metaphysics.

Geometric axioms are neither synthetic a priori nor empirical. They are
more properly understood as definitions.Thus
when one set of axioms is preferred over another the selection is a matter of
convention.Poincareís philosophy
of science was formed by his approach to mathematics which was broadly
geometric.It is governed by the
criteria of simplicity of expression rather than by which geometry is ultimately
correct. A sketch of Kantís theory of knowledge that defined the existence of
mathematical truths a central pillar to his philosophy.In particular, he rests support on the truths of Euclidean geometry. His
inability to realize at that time the existence of any other geometry convinced
him that it was the only one.Thereby,the truths demonstrated by Euclidean systems and the existence of a
priori synthetic propositions were a guarantee.The discovery of non-Euclidean geometry opened other variables for
Kantís arguments.That Euclidean
geometry is used to describe the motion of bodies in space, it makes no sense to
ask if physical space is really Euclidean.Discovery in mathematics is similar to the discovery in the physical
sciences whereas the former is a construction of the human mind.The latter must be considered as an order of nature that is independent
of mind.Newton became disenchanted
with his original version of calculus and that of Leibniz and around 1680 had
proceeded to develop a third version of calculus based on geometry.This geometric calculus is the mathematical engine behind Newtonís Principia.

Conventionalism as geometrical and mathematical truths are created by our
choices, not dictated by or imposed on us by scientific theory.The idea that geometrical truth is truth we create by the understanding
of certain conventions in the discovery of non-Euclidean geometries.Subsequent to this discovery,Euclidean
geometries had been considered as a paradigm of a priori knowledge.The further discovery of alternative systems of geometry are consistent
with making Euclidean geometry seem dismissed without interfering with
rationality.Whether we utilize the
Euclidean system or non-Euclidean system seems to be a matter of choice founded
on pragmatic considerations such as simplicity and convenience.

The Euclidean, Lobachevsky-Bolyai-Gauss, and Riemannian geometries are
united in the same space, by the
Smarandache Geometries, 1969.A
Smarandache Geometry is a geometry in which at least one axiom behaves
differently (validated and invalidated in many ways) in the same space.These geometries are, therefore partially Euclidean and partially
Non-Euclidean.The geometries in
their importance unite and generalize all together and separate them as well.Hilbertís relations of incidence, betweenness, and congruence are made
clearer through the negations ofSmarandacheís
Anti-Geometry.Smarandacheís
geometries fall under the following categories: Paradoxist Geometry,
Non-Geometry, Counter-Projective Geometry, and Anti-Geometry.It seems that Smarandache Geometries are connected with the
Theory of Relativity (because they include the Riemannian geometry in a
subspace) and with the (adjacent) Parallel Universes.

Science provides a fruitful way of expressing the relationships between
types or sets of sensations, enabling reliable predictions to be offered.These sensations of sets of data reflect the world that causes them or
causal determination; as a limited objectivity of science that derives from this
fact,but science does not suppose
to determine the nature of that underlying world.It is the underlying structure found through geometry that has driven the
world of geometers to artistic expressions.Geometrical art can through conventions and choices which are
determinable by rule may appear to be empirical, but are in fact postulates that
geometers have chosen to select as implicit definitions.The choice to select a particular curve to represent a finite set of
points requires a judgment as to that which is simpler.There are theories which can be drawn that lead to postulate underlying
entities or structures.These
abstract entities or models may seem explanatory, but strictly speaking are no
more than visual devices useful for calculation.

Abstract entities,are
sometimes collected under universal categories,that include mathematical objects, such as numbers, sets, and geometrical
figures, propositions, and relations.Abstracta,are stated to be abstracted from particulars.The abstract square or triangle have only the properties common to all
squares or triangles, and none peculiar to any particular square or triangle;that they have not particular color, size, or specific type whereby they
may be used for an artistic purpose.Abstracta
are admitted to an ontology by Quineís criterion if they must exist in order
to make the mechanics of the structure to be real and true.Properties and relations may be needed to account for resemblance among
particulars, such as the blueness shared amongst all blue things.

Concrete intuition and understanding is a major role in the appreciation
of geometry as intersections both in art and science.This bares great value not only to the participating geometer artists but
to the scholars for their research.In
the presentation of geometry,we
can bridge visual intuitive aspects with visual imagination.In this statement,I have outlined for geometry and art without strict
definitions of concepts or with any actual computations.Thus,the presentation of
geometry as a brushstroke to approach visual intuition should give a much
broader range of appreciation to mathematics.

Clifford Singer, 2001 ”

References:

David Papineau (editor), (1999),
The Philosophy of Science, Oxford
Readings in Philosophy.

Morris Kline, (1953), Mathematics
In Western Culture, Oxford University Press.